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A Course Material on
Electrical machines I
By
Mrs. A.Saranya
Assistant PROFESSOR
DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING
SASURIE COLLEGE OF ENGINEERING
VIJAYAMANGALAM 638 056
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QUALITY CERTIFICATE
This is to certify that the e-course material
Subject Code : EE 2251
Scubject : Electrical Machines I
Class : II Year EEE
being prepared by me and it meets the knowledge requirement of
the university
curriculum.
Signature of the Author
Name:
Designation:
This is to certify that the course material being prepared by
Mrs A.Saranya is of
adequate quality. She has referred more than five books among
them minimum one is
from aboard author.
Signature of HD
Name: S.SRIRAM
SEAL
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S.NO CONTENT PAGE NO
CHAPTER I MAGNETIC CIRCUITS AND MAGNETIC
MATERIAL
1.1 Introduction 8
1.2 Magnetic Circuits 8
1.2.1 Magnetic Material 10
1.2.2Magnetic Effect By Electric Current 11
1.3 Laws Governing Magnetic Circuits 11
1.3.1. Magnetic flux 11
1.3.2. Magnetic field strength 11
1.3.3.Flux density 12
1.3.4.Magneto-Motive Force 12
1.3.5.Magnetic Reluctance 12
1.3.6. Residual Magnetism 12
1.3.7. Magnetic Saturation 12
1.3.8. End Rule 12
1.3.9. Lenzs Law 13
1.3.10. Electro magnetic induction 13
1.3.11. Fleming's Right Hand Rule 13
1.4 Flux Linkage, Inductance and Energy 13
1.4.1. Flux Linkage 13
1.4.2 Inductance and Energy 14
1.5 Statically And Dynamically Induced Emf 14
1.5.1 Statically Induced Emf 14
1.5.1.1 Self Inductucedemf 14
1.5.1.2 Mutually Induced EMF 15
1.5.2 Dynamically induced EMF 15
1.6 Properties of Magnetic Materials 15
1.6.1 Magnetic Hysteresis 15
1.6.2 Hysteresis Loop 16
1.7 Iron or Core losses 16
1.7.1. Hysteresis loss 16
1.7.2 Eddy current loss 17
1.7.3 Mechanical losses 18
1.8 Ac Operation Of Magnetic Circuits 18
1.9 Transformer As A Magnetically Coupled Circuit 19
1.10 Solved problems 21
CHAPTER 2 TRANSFORMER
2.1 Principle Of Operation 26
2.1.1 Basic Principle 26
2.1.2 An ideal Transformer 32
2.1.3 Induction Law 32
2.2 Equivalent Circuit 34
2.3 Transformer Losses 35
2.4 Transformer Tests 37
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2.4.1 Open-circuit or No-load Test 37
2.4.2 Short-circuit or Impedance Test. 38
2.5 Efficiency 39
2.6 Voltage Regulation 41
2.6.1 Circuit Diagram 42
2.6.2 Procedure 42
2.6.3 Observation Table 42
2.6.4 Calculation 42
2.6.5 Discussion 42
2.7 Auto Transformer 43
2.8 Three-phase autotransformer connection 44
2.8.1 Design, Vector group 44
2.8.2 Three-Leg Transformer 45
2.9 Parallel Operation Of Transformers 46
2.10 Tap Changing 48
SOLVED PROBLEMS 52
CHAPTER- 3 ELECTROMECHANICAL ENERGY CONVERSION
AND CONCEPTS INROTATING MACHINES
3.1 Energy In Magnetic Systems 58
3.1.1Electromechanical-Energy-Conversion Principles 58
3.1.2Forces and Torques in Magnetic Field Systems 58
3.2 The Field Energy 59
3.2.1 Energy Balance 59
3.3 The Co Energy 60
3.4 Force In A Singly Excited Magnetic Field System 62
3.4.1 Model & Analysis 62
3.5 Force In A Multiply Excited Magnetic Field System 64
3.6 MMf Of Distributed Windings 67
3.6.1 Alternating Field Distribution 67
3.6.2 Rotating field 68
3.6.3 Three-phase winding 68
3.6.4 Determination of slot mmf for different moments
(temporal)
69
3.7 Magnetic Fields In Rotating Machines 69
3.7.1 Winding factor 69
3.7.2 Distribution factor 72
3.7.3 Pitch factor 73
3.8 Rotating Mmf Waves 73
3.8.1 Voltage induction caused by influence of rotating field
74
3.8.2 Flux linkage 75
3.8.3 Induced voltage, slip 75
3.9 Torque In Ac And Dc Machines 76
SOLVED PROBLEMS 78
CHAPTER 4 DC GENERATOR
4.1 Principles Of D.C. Machines 83
4.2 Construction of DC. Machines 83
4.2.1 Frame 83
4.2.2 Yoke 83
4.2.3 End Shields or Bearings 84
4.2.4 Main poles 84
4.2.5 Armature 84
4.2.6 Commutator 85
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4.3 Lap Winding 86
4.4 Wave Winding 87
4.5 EMF Equation 88
4.6 Armature reaction 89
4.7 Methods Of Excitation 90
4.8 Commutation And Interpoles 91
4.9 Generator Characteristics 92
4.9.1.OpenCircuitCharacteristic(O.C.C. 92
4.9.2. Internal or Total characteristic (E/Ia) 92
4.9.3. External Characteristic (V/IL) 92
4.9.4. No-load Saturation Characteristic (E0/If) 93
4.9.5.Separately-Excited Generator 93
4.9.6. External Characteristic (V/I) 94
CHAPTER 5 DC MOTORS
5.1 Principle 96
5.2 Operation 96
5.3 Types of DC motor 96
5.4 Motor Characteristics 98
5.4.1 Torque/Speed Curves 98
5.4.2 Power/Torque And Power/Speed Curves 100
5.5 Speed Control Of Dc Shunt Motor 101
5.5.1Speed Control by Varying Armature Resistance 101
5.5.2 Speed Control by Varying Field Current 102
5.6 Starting Of Dc Motors 103
5.7 Three Point Starter 104
5.8 Four-Point Starter 106
5.9 Swinburnes Test 106 5.10 Hopkinsons test 109
SOLVED PROBLEMS 115
GLOSSARY 118
TWO MARK QUESTION WITH ANSWER 120
UNIVERSITY QUESTION BANK 129
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AIM
To expose the students to the basic principles of Electro
mechanical Energy
Conversion in Electrical Apparatus and the operation of
Transformers and DC
Machines.
OBJECTIVES:
To introduce techniques of magnetic-circuit analysis and
introduce magnetic materials
To familiarize the constructional details, the principle of
operation, prediction of performance, the methods of testing the
transformers and three
phase transformer connections.
To study the working principles of electrical machines using the
concepts of electromechanical energy conversion principles and
derive expressions for
generated voltage and torque developed in all Electrical
Machines.
To study the working principles of DC machines as Generator
types, determination of their no load / load characteristics,
starting and methods of
speed control of motors.
To estimate the various losses taking place in D.C. Motor and to
study the different testing methods to arrive at their
performance.
UNIT I MAGNETIC CIRCUITS AND MAGNETIC MATERIALS
Magnetic circuits Laws governing magnetic circuits - Flux
linkage, Inductance and energy Statically and Dynamically induced
EMF - Torque Properties of magnetic materials, Hysterisis and Eddy
Current losses - AC excitation,
introduction to permanent magnets-Transformer as a magnetically
coupled circuit..
UNIT II TRANSFORMERS 9
Construction principle of operation equivalent circuit
parameters phasor diagrams, losses testing efficiency and voltage
regulation-all day efficiency-Sumpners test, per unit
representation inrush current - three phase
transformers-connections Scott Connection Phasing of transformer
parallel operation of three phase transformers-auto transformer tap
changing transformers- tertiary Winding UNIT III ELECTROMECHANICAL
ENERGY CONVERSION AND
CONCEPTS IN ROTATING MACHINES
Energy in magnetic systems field energy, co energy and
mechanical force singly and multiply excited systems. Energy in
magnetic system Field energy and coenergy -force and torque
equations singly and multiply excited magnetic field systems-mmf of
distributed windings Winding Inductances-, magnetic fields in
rotating machines rotating mmf waves magnetic saturation and
leakage fluxes. UNIT IV DC GENERATORS
Construction and components of DC Machine Principle of operation
- Lap and wave windings-EMF equations circuit model armature
reaction methods of excitation-commutation and inter poles -
compensating winding characteristics of DC generators.
UNIT V DC MOTORS 9
Principle and operations - types of DC Motors Speed Torque
Characteristics of DC Motors-starting and speed control of DC
motors Plugging, dynamic and regenerative braking- testing and
efficiency Retardation test- Swinburnes test and Hopkinsons test -
Permanent magnet dc motors(PMDC)-DC Motor applications
TEXT BOOKS
1. Nagrath I. J and Kothari D. P. Electric Machines, Tata McGraw
Hill Publishing Company Ltd,1990.
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2. P.S. Bimbhra, Electrical Machinery, Khanna Publishers, 2003.
REFERENCES
1. Fitzgerald.A.E., Charles KingselyJr, Stephen D.Umans,
Electric Machinery, McGraw Hill BooksCompany, 1992.
2. P. C. Sen., Principles of Electrical Machines and Power
Electronics, John Wiley&Sons, 1997.
3. K. Murugesh Kumar, Electric Machines, Vikas publishing house
Pvt Ltd, 2002.
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CHAPTER- 1
MAGNETIC CIRCUITS AND MAGNETIC MATERIALS
1.1 Introduction
The law of conservation of energy states that the energy cannot
be related or
destroyed but it can be converted from one form to other. An
electrical energy does
not occur naturally and also cannot be stored. Hence the efforts
are made to generate it
continuously to meet the large demands. But to generate an
electrical energy means to
convert some other form of energy into an electrical form,
according to law of
conservation of energy. A commonly used method to generate an
electrical energy is
converting mechanical energy into electrical with the help of a
rotating device. Such a
machine which converts the mechanical energy into an electrical
energy is called a
generator. The input mechanical energy can be achieved from
steam turbines, steam
engines or using potential energy of water to run hydraulic
turbines. Such a device
which inputs a mechanical energy to a generator is called a
prime mover. While
converting energy from mechanical to electrical form, some
losses take place. The
losses are kept to minimum value by properly designing the
machine. Practically the
efficiencies of large generators are above 90 %
1.2 Magnetic Circuits
In a magnetic circuit, the magnetic lines of force leaves the
north poles passes
through the entire circuit and return the starting point. A
magnetic circuit usually
consist of materials having high permeability such as iron ,
soft steel etc., These
materials offer very small opposition to the flow of magnetic
flux . consider a coil of N
turns would on an iron core
Amperes law
. . : magnetic field intensity vector, : current density.C S
H dl J da H J
. 0 : magnetic flux density vector.S
B da B magnetic flux density is conserved
0
7
0 0
: magnetic permeability of medium.
: permeability of free space =4 10
: relative permeability
r
r
B H
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. .C S
Ni FH dl J da : magnetomotive force (mmf, ampere-turns).
Magnetic flux crossing surface S: .S
B da (Weber, Wb)
: flux in core,
: flux density in the core
: cross-sectional area of the core.
c c c c
c
c
B A
B
A
.
: reluctance
cc c c c
cC
c
c
BH l l Ni F l F
A
lF
A
H dl
Fig. 1.2 Magnetic circuit with air gap.
Flux is the same in the magnetic core and the air-gap.
flux density in the magnetic core.
flux density in the air-gap.
c
c
g
g
BA
BA
0 0
mmf .
( )
: reluctance of core, : reluctance of air-gap.
gc cc c g c
c gC
c g
c g
c g
BB l gH l H g Ni F F l g
A A
FF
H dl
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Fig 1.1 Analogy between electric and magnetic circuits.
.
magnetism plays an important role in electricity. Electrical
appliances like Generator,
Motor, Measuring instruments and Transformer are based on the
electromagnetic
principle and also the important components of Television, Radio
and Aero plane are
working on the same principle.
1.2.1 Magnetic Material
Magnetic materials are classified based on the property called
permeability as
1. Dia Magnetic Materials
2. Para Magnetic Materials
3. Ferro Magnetic Materials
1. Dia Magnetic Materials
The materials whose permeability is below unity are called Dia
magnetic
materials. They are repelled by magnet.
Ex. Lead, gold, copper, glass, mercury
2. Para Magnetic Materials
The materials with permeability above unity are called Para
magnetic materials. The
force of attraction by a magnet towards these materials is
low.
Ex.: Copper Sulphate, Oxygen, Platinum, Aluminum.
3. Ferro Magnetic Materials
The materials with permeability thousands of times more than
that of
paramagnetic materials are called Ferro magnetic materials. They
are very much
attracted by the magnet.
Ex. Iron, Cobalt, Nickel.
Permanent Magnet
Permanent magnet means, the magnetic materials which will retain
the
magnetic property at a] l times permanently. This type of
magnets is manufactured by
aluminum, nickel, iron, cobalt steel (ALNICO).
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To make a permanent magnet a coil is wound over a magnetic
material and DC
supply is passed through the coil.
Electro Magnet
Insulated wire wound on a bobbin in many turns and layers in
which current is
flowing and a soft iron piece placed in the bobbin is called
electromagnet.
Figure 1.2 This is used in all electrical machines,
transformers, electric bells. It is also used in
a machine used by doctors to pull out iron filing from eyes,
etc.
1.2.2 Magnetic Effect By Electric Current
If current passes through a conductor magnetic field is set up
around the
conductor. The quantity of the magnetic field is proportion to
the current. The
direction of the magnetic field is found by right hand rule or
max well's corkscrew
rule. Magnetic Flux The magnetic flux in a magnetic circuit is
equal to the total
number of lines existing on the cross-section of the magnetic
core at right angle to
the direction of the flux.
H=
Where,
- total flux
N - number of turns
I - current in amperes
S - reluctance
- permeability of free space
0 - relative permeability
a - magnetic path cross-sectional area in m2
l - lengh of magnetic path in metres
1.3 Laws Governing Magnetic Circuits
1.3.1. Magnetic flux:
The magnetic lines of force produced by a magnet is called
magnetic flux. It is
denoted by and its unit is Weber.
1.3.2. Magnetic field strength
This is also known as field intensity, magnetic intensity or
magnetic field, and is
represented by the letter H. Its unit is ampere turns per
metre.
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H=
1.3.3.Flux density
The total number of lines of force per square metre of the
cross-sectional area of the
magnetic core is called flux density, and is represented by the
symbol B. Its SI unit (in
the MKS system) is testa (weber per metre square).
B=
where
-total flux in webers
A - area of the core in square metres
B - flux density in weber/metre square.
1.3.4 .Magneto-Motive Force
The amount of flux density setup in the core is dependent upon
five factors - the
current, number of turns, material of the magnetic core, length
of core and the cross-
sectional area of the core. More current and the more turns of
wire we use, the greater
will be the magnetizing effect. We call this product of the
turns and current the magneto
motive force (mmf), similar to the electromotive force
(ernf).
MMF = NI ampere - turns
Where mmf is the magneto motive force in ampere turns
N is the number of turns, A.
1.3.5.Magnetic Reluctance
In the magnetic circuit there is something analogous to
electrical resistance, and is
called reluctance, (symbol S). The total flux is inversely
proportional to the reluctance and
so if we denote mmf by ampere turns. we can write
S=
Where, S - reluctance
I - length of the magnetic path in meters
o- permeability of free space r - relative permeability
a - cross-sectional area
1.3.6. Residual Magnetism
It is the magnetism which remains in a material when the
effective magnetizing
force has been reduced to zero.
1.3.7. Magnetic Saturation
The limit beyond which the strength of a magnet cannot be
increased is called
magnetic saturation.
1.3.8. End Rule
According to this rule the current direction when looked from
one end of the coil
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is in clock wise direction then that end is South Pole. If the
current direction is in anti
clock wise direction then that end is North Pole.
1.3.9. Lens Law
When an emf is induced in a circuit electromagnetically the
current set up always opposes
the motion or change in current which produces it.
1.3.10. Electro magnetic induction
Electromagnetic induction means the electricity induced by the
magnetic field
Faraday's Laws of Electro Magnetic Induction
There are two laws of Faraday's laws of electromagnetic
induction. They are,
1) First Law 2) Second Law
First Law
Whenever a conductor cuts the magnetic flux lines an emf is
induced in the conductor.
Second Law
The magnitude of the induced emf is equal to the rate of change
of flux-linkages.
1.3.11. Fleming's Right Hand Rule
This rule is used to find out the direction of dynamically
induced emf. According to
the rule hold out the right hand with the Index finger middle
finger and thumb at the
right angels to each others. If the index finger represents the
direction of the lines of
flux, the thumb points in the direction of motion then middle
finger points in the
direction of induced current.
Figure 1.3 Fleming's Right Hand Rule
1.4 Flux Linkage, Inductance and Energy
1.4.1. Flux Linkage
When flux is changing with time and relative motion between the
coils flux exist
between both the coils or conductors and emf induces in both
coil and the total induced
emf e is given as
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1.4.2 Inductance and Energy
A coil wound on a magnetic core, is used frequently used in
electric circuits.
The coil may be representsd by an ideal circuit element called
inductance which is
defined as the flux linkage of the coil per ampere of its
circuit
1.5. Statically And Dynamically Induced Emf.
Induced electro motive forces are of two types. They are,
i) Dynamically induced emf.
ii) Statically induced emf .
1.5.1 Statically Induced Emf
Statically Induced emf is of two types. They are
1 .Self induced emf
2. Mutually induced emf.
1.5.1.1 Self Inductuced emf
Self induction is that phenomenon where by a change in the
current in a conductor
induces an emf in the conductor itself. i.e. when a conductor is
given current, flux will be
produced, and if the current is changed the flux also changes,
as per Faraday's law when
there is a change of flux, an emf will be induced. This is
called self induction. The induced
emf will be always opposite in direction to the applied emf. The
opposing emf thus
produced is called the counter emf of self induction.
Uses of Self induction
.1. In the fluorescent tubes for starting purpose and to reduce
the voltage.
2. In regulators, to give reduced voltage to the fans.
3. In lightning arrester.
4. In auto- transformers.
5. In smooth choke which is used in welding plant.
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1.5.1.2 Mutually Induced EMF
It is the electromagnetic induction produced by one circuit in
the near by second circuits
due to the variable flux of the first circuit cutting the
conductor of the second circuit, that
means when two coils or circuits are kept near to each other and
if current is given to one
circuit and it is changed, the flux produced due to that current
which is linking both the
coils or circuits cuts both the coils, an emf will be produced
in both the circuits. The
production of emf in second coil is due to the variation of
current in first coil known
as mutual induction.
Uses:
1. It is used in ignition coil which is used in motor car.
2. It is also used in inductance furnace.
3. It is used for the principle of transformer
1.5.2 Dynamically induced EMF
Dynamically induced emf means an emf induced in a conductor when
the conductor
moves across a magnetic field. The Figure shows when a conductor
Awith the length L moves across a B wb/m2.
Figure1.4 Dynamically induced emf.
Flux density with V velocity, then the dynamically induced emf
is induced in the conductor. This induced emf is utilized in the
generator. The quantity of the emf can be
calculated using the equation
emf= Blv volt
1.6. Properties of Magnetic Materials
1.6.1 Magnetic Hysteresis
It may be defined as the lagging of magnetization or Induction
flux density (B) behind the
magnetizing force (H). It may also be defined as a quality of a
magnetic substance due
to which energy is dissipated in it on the reversal of its
magnetism
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Fig 1.5 Magnetic Hysteresis loop
1.6.2 Hysteresis Loop
Let us take a un magnetized bar of iron AB and magnetize in by
placing it within
the magnetizing field of a solenoid (H). The Field can be
increased or decreased by
increasing or decreasing current through it. Let `H' be
increased in step from zero up to
a certain maximum value and the corresponding of induction flux
density (B) is noted. If
we plot the relation between H and B, a curve like OA, as shown
in Figure, is obtained.
The material becomes magnetically saturated at H = OM and has,
at that time, a
maximum flux density, established through it. If H is now
decreased gradually (by
decreasing solenoid current) flux density B will not decrease
along AO (as might be
expected) but will decrease less rapidly along AC. When it is
Zero B is not zero, but has
a definite value = OC. It means that on removing the magnetizing
force H, the iron bar
is not completely demagnetized. This value of B (=OC) is called
the residual flux
density.
To demagnetize the iron bar we have to apply the magnetizing
force
H in the reverse direction. When H is reversed by reversing
current through the solenoid,
then B is reduced to Zero at point D where H - OD. This value of
H required to wipe off
residual magnetism is known as coercive force and is a measure
of the coercivity of
materials i.e. its `tenacity' with which it holds on to its
magnetism. After the
magnetization has been reduced to zero value of H is further
increased in the negative i.e.
reverse direction, the iron bar again reaches a state of
magnetic saturation represented
by point E. By taking H back from its value corresponding to
negative saturation (=OL) to
its value for positive saturation (=OM), a similar curve EFGA is
obtained. If we again start
from G, the same curve GACDEFG is obtained once again. It is
seen that B always lags
behind H the two never attain zero value simultaneously. This
lagging of B, behind H is
given the name Hysteresis' which literally means `to lag
behind.' The closed Loop
ACDEFGA, which is obtained when iron bar is taken through one
complete cycle of
reversal of magnetization, is known as Hysteresis loop.
1.7. Iron or Core losses
These losses occur in the armature of a d.c. machine and are due
to the rotation
of armature in the magnetic field of the poles.
They are of two types
(i) hysteresis loss (ii) (ii) eddy current loss.
1.7.1. Hysteresis loss
Hysteresis loss occurs in the armature of the d.c. machine since
any given part
of the armature is subjected to magnetic field reversals as it
passes under successive
poles.Figure. (1.36) shows an armature rotating in two-pole
machine. Consider a small
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piece ab of the armature. When the piece ab is under N-pole, the
magnetic lines pass
from a to b. Half arevolution later, the same piece of iron is
under S-pole and magnetic
lines pass from b to a so that magnetism in the iron is
reversed. In order to reverse
continuously the molecular magnets in the armature core, some
amount of power has
to be spent which is called hysteresis loss. It is given by
Steinmetz formula. This
formula is Hysteresis loss,
Ph=B16
maxfV watts
where Bmax = Maximum flux density in armature
f = Frequency of magnetic reversals
V = Volume of armature in m3
h = Steinmetz hysteresis co-efficient
Figure 1.6 Hysteresis loss In order to reduce this loss in a
d.c. machine, armature core is made of such materials
which have a low value of Steinmetz hysteresis co-efficient
e.g., silicon steel.
1.7.2 Eddy current loss
In addition to the voltages induced in the armature conductors,
there are also
voltages induced in the armature core. These voltages produce
circulating currents in
the armature core as shown in Figure. (1.37). These are called
eddy currents and power
loss due to their flow is called eddy current loss. The eddy
current loss appears as heat
which raises the temperature of the machine and
lowers its efficiency. If a continuous solid iron core is used,
the resistance to eddy
current path will be small due to large cross-sectional area of
the core. Consequently,
the magnitude of eddy current and hence eddy current loss will
be large. The
magnitude of eddy current can be reduced by making core
resistance as high as
practical. The core resistance can be greatly increased by
constructing the core of thin,
round iron sheets called laminations.The laminations are
insulated from each other
with a coating of varnish. The insulating coating has a high
resistance, so very little
current flows from one lamination to the other. Also, because
each lamination is very
thin, the resistance to current flowing through the width of a
lamination is also quite
large. Thus laminating a core increases the core resistance
which decreases the eddy
current and hence the eddy current loss.
Eddy current loss, Pe = KeB2
maxf2t2V watts
where ,
Ke = Constant
Bmax = Maximum flux density in Wb/m2
f = Frequency of magnetic reversals in Hz
t = Thickness of lamination in m
V = Volume of core in m3
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Figure 1.7 Eddy current loss
It may be noted that eddy current loss depends upon the square
of lamination
thickness. For this reason, lamination thickness should be kept
as small as possible.
1.7.3 Mechanical losses
These losses are due to friction and windage.
(i) friction loss e.g., bearing friction, brush friction
etc.
(ii) windage loss i.e., air friction of rotating armature.
These losses depend upon the speed of the machine. But for a
given speed, they are
practically constant.
Note. Iron losses and mechanical losses together are called
stray losses
Eddy current
When the armature with conductors rotates in the magnetic field
and cuts the magnetic
lines, an emf will be induced in the conductors. As the armature
is made of a metal and
metal being a conductor, emf will be induced in that metal also
and circulate the current
called eddy current. These current produces some effects which
can be utilized. This
current are also called as Focault current. Methods of
Minimizing Eddy current always
tends to flow at the right angles to the direction ofthe flux,
if the resistance of the path is
increased by laminating the cores. The power loss can be reduced
because the eddy
current loss varies as the square of the thickness of the
laminations.
Figure 1.8 Eddy current
1.8 Ac Operation Of Magnetic Circuits
For establishing a magnetic field, energy must be spent, though
to energy is
required to maintain it. Take the example of the exciting coils
of an electromagnet. The
energy supplied to it is spent in two ways, (i) Part of it goes
to meet I2R loss and is lost
once for all (ii) part of it goes to create flux and is stored
in the magnetic field as
potential energy, and is similar to the potential energy of a
raised weight, when a mass M
is raised through a height of H, the potential energy stored in
it is mgh. Work is done in
raising this mass, but once raised to a certain height. No
further expenditure of energy
is required to maintain it at that position. This mechanical
potential energy can be
recovered so can be electric energy stored in a magnetic field.
When current through an
inductive coil is gradually changed from Zero to a maximum,
value then every change
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of it is opposed by the self-induced emf. Produced due to this
change. Energy is needed
to overcome this opposition. This energy is stored in the
magnetic field of the coil and is,
later on, recovered when those field collapse.
In many applications and machines such as transformer and a.c
machines, the
magnetic circuits are excited by a.c supply. In such an
operation, Inductance plays
vital role even in steady state operation though in d.c it acts
as a short circuit. In such a
case the flux is determined by the a.c voltage applied and the
frequency, thus the
exciting current has to adjust itself according to the flux so
that every time B-H
relationship is satisfied.
Consider a coil having N turns wound on iron core as shown in
fig
The coil carries an alternating current i varying sinusoidally.
Thus the flux
produced by the exciting current I is also sinusoidally varying
with time.
According to Faradays law as flux changes with respect to coli,
the e.m.f gets induced in the coil given by,
e= N = N
Em = Maximum value = N
E= r.m.s value = =
E= = 4.44 fN
But = Ac Bm
The sign of e.m.f induced must be determined according to lens
law, opposing the changes in the flux. The current and flux are in
phase as current produces flux
instantaneously. Now induced e.m.f is cosine term and thus leads
the flux and current
by .this is called back e.m.f as it opposes the applied voltage.
The resistance drops
is very small and is neglecte3d in most of the electromagnetic
devices
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1.9. Transformer As A Magnetically Coupled Circuit
A two winding transformer where R1 and R2 are the primary and
secondary winding
resistance. The primary current i1 into the dotted terminal
produces
Core flux = 21
Leakage flux = 1
Total flux = 1 + 21
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1.10 Solved problems
Eg .No.1
A magnetic circuit with a single air gap is shown in Fig. 1.24.
The core dimensions
are:
Cross-sectional area Ac = 1.8 10-3
m2
Mean core length lc = 0.6 m
Gap length g = 2.3 x 10-3
m
N = 83 turns
Assume that the core is of infinite permeability ( ) and neglect
the effects of
fringing fields at the air gap and leakage flux. (a) Calculate
the reluctance of the core
cR and that of the gap gR
. For a current of i = 1.5 A, calculate (b) the total flux ,
(c)
the flux linkages of the coil, and (d) the coil inductance
L.
Solution:
0 since cR
36
7 3
0
2.3 101.017 10 A/Wb
4 10 1.8 10g
c
gR
A
4
6
83 1.51.224 10 Wb
1.017 10c g
Ni
R R
21.016 10 WbN
21.016 106.773 mH
1.5L
i
Eg .No.2
Consider the magnetic circuit of with the dimensions of Problem
1.1. Assuming
infinite core permeability, calculate (a) the number of turns
required to achieve an
inductance of 12 mH and (b) the inductor current which will
result in a core flux
density of 1.0 T.
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voltage
T
e
t
max
E
E
max
Solution:
23 3 612 10 mH 12 10 1.017 10 110.47 110 turns
g
NL N N
R
3
3
3
1.0 T 1.8 10 Wb
110 1.8 1016.5 A
12 10
c g g cB B B A
Ni
L L
Eg .No.3
A square voltage wave having a fundamental frequency of 60 Hz
and equal positive
and negative half cycles of amplitude E is applied to a
1000-turn winding surrounding
a closed iron core of 1.25 x 10-3
m2 cross section. Neglect both the winding resistance
and any effects of leakage flux.
(a) Sketch the voltage, the winding flux linkage, and the core
flux as a function
of time.
(b) Find the maximum permissible value of E if the maximum flux
density is
not to exceed 1.15 T.
max maxmax max max
3
( )( ) ( ). 4 4 4
/ 2
4 60 1000 1.25 10 1.15 345 V
c
de t e t dt E f fN fNA B
dt T
E
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Eg.No.4
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Eg.no.5
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CHAPTER- 2
TRANSFORMER
2.1 Principle Of Operation
A transformer is a device that transfers electrical energy from
one circuit to
another through inductively coupled conductor. A varying current
in the first or
primary winding creates a varying magnetic flux in the
transformer core, and thus a
varying magnetic field through the secondary winding. This
varying magnetic field
induces a varying electromotive force EMF or voltage in the
secondary winding. This
effect is called mutual induction.
If a load is connected to the secondary, an electric current
will flow in the
secondary winding and electrical energy will be transferred from
the primary circuit
through the transformer to the load. In an ideal transformer,
the induced voltage in the
secondary winding is in proportion to the primary voltage , and
is given by the ratio of
the number of turns in the secondary to the number of turns in
the primary as follows:
By appropriate selection of the ratio of turns, a transformer
thus allows an
alternating current (AC) voltage to be "stepped up" by making
greater than , or
"stepped down" by making less than .
2.1.1 Basic Principle
Construction
Figure 2.1 Laminated core transformer showing edge of
laminations
Laminated steel cores
Transformer use at power or audio frequencies typically have
cores made of
high permeability Si steel. The steel has permeability many
times that of free and the
core thus serves to greatly reduce the magnetizing current and
confine the flux to a
path which closely couples the windings. Early transformer
developers soon realized
that cores constructed from solid iron resulted in prohibitive
eddy-current losses, and
their designs mitigated this effect with cores consisting of
bundles of insulated iron
wires. Later designs constructed the core by stacking layers of
thin steel laminations, a
principle that has remained in use. Each lamination is insulated
from its neighbors by a
thin non-conducting layer of insulation. The universal
transformer equation indicates a
minimum cross-sectional area for the core to avoid
saturation.
The effect of laminations is to confine eddy currents to highly
elliptical paths
that enclose little flux, and so reduce their magnitude. Thinner
laminations reduce
losses, but are more laborious and expensive to construct. Thin
laminations are
generally used on high frequency transformers, with some types
of very thin steel
laminations able to operate up to 10 kHz.
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Figure 2.2laminating the core greatly reduces eddy-current
losses
One common design of laminated core is made from interleaved
stacks of E-
shaped steel sheets capped with shaped pieces, leading to its
name of "E-I
transformer. Such a design tends to exhibit more losses, but is
very economical to manufacture. The cut-core or C-core type is made
by winding a steel strip around a
rectangular form and then bonding the layers together. It is
then cut in two, forming
two C shapes, and the core assembled by binding the two C halves
together with a
steel strap.[73]
They have the advantage that the flux is always oriented
parallel to the
metal grains, reducing reluctance.
A steel core's permanence means that it retains a static
magnetic field when
power is removed. When power is then reapplied, the residual
field will cause a high
inrush until the effect of the remaining magnetism is reduced,
usually after a few
cycles of the applied alternating current. Over current
protection devices such
as fuses must be selected to allow this harmless inrush to pass.
On transformers
connected to long, overhead power transmission lines, induced
currents due
to geomagnetic disturbances during solar storms can cause
saturation of the core and
operation of transformer protection devices.
Distribution transformers can achieve low no-load losses by
using cores made with
low-loss high-permeability silicon steel or amorphous
(non-crystalline) metal alloy.
The higher initial cost of the core material is offset over the
life of the transformer by
its lower losses at light load.
Solid cores
Powdered iron cores are used in circuits such as switch-mode
power supplies
that operate above mains frequencies and up to a few tens of
kilohertz. These materials
combine high magneticpermeancehigh bulk electrical resistivity.
For frequencies
extending beyond the VHF band, cores made from
non-conductive
magnetic ceramic materials called ferrites are common. Some
radio-frequency
transformers also have movable cores (sometimes called 'slugs')
which allow
adjustment of the coupling coefficient (and bandwidth) of tuned
radio-frequency
circuits.
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Toroidal cores
Figure 2.3 Small toroidal core transformer
Toroidal transformers are built around a ring-shaped core,
which, depending on
operating frequency, is made from a long strip of silicon steel
or perm alloy wound
into a coil, powdered iron, or ferrite. A strip construction
ensures that the grain
boundaries are optimally aligned, improving the transformer's
efficiency by reducing
the core's reluctance. The closed ring shape eliminates air gaps
inherent in the
construction of an E-I core.[78]
The cross-section of the ring is usually square or
rectangular, but more expensive cores with circular
cross-sections are also available.
The primary and secondary coils are often wound concentrically
to cover the entire
surface of the core. This minimizes the length of wire needed,
and also provides
screening to minimize the core's magnetic field from generating
electromagnetic.
Toroidal transformers are more efficient than the cheaper
laminated E-I types
for a similar power level. Other advantages compared to E-I
types, include smaller size
(about half), lower weight (about half), less mechanical hum
(making them superior in
audio amplifiers), lower exterior magnetic field (about one
tenth), low off-load losses
(making them more efficient in standby circuits), single-bolt
mounting, and greater
choice of shapes. The main disadvantages are higher cost and
limited power capacity
(see "Classification" above). Because of the lack of a residual
gap in the magnetic
path, toroidal transformers also tend to exhibit higher inrush
current, compared to
laminated E-I types.
Ferrite toroidal cores are used at higher frequencies, typically
between a few
tens of kilohertz to hundreds of megahertz, to reduce losses,
physical size, and weight
of a switched-mode power supply. A drawback of toroidal
transformer construction is
the higher labor cost of winding. This is because it is
necessary to pass the entire
length of a coil winding through the core aperture each time a
single turn is added to
the coil. As a consequence, toroidal transformers are uncommon
above ratings of a
few kVA. Small distribution transformers may achieve some of the
benefits of a
toroidal core by splitting it and forcing it open, then
inserting a bobbin containing
primary and secondary windings.
Air cores
A physical core is not an absolute requisite and a functioning
transformer can
be produced simply by placing the windings near each other, an
arrangement termed
an "air-core" transformer. The air which comprises the magnetic
circuit is essentially
lossless, and so an air-core transformer eliminates loss due to
hysteresis in the core
material.[41]
The leakage inductance is inevitably high, resulting in very
poor
regulation, and so such designs are unsuitable for use in power
distribution. They have
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however very high bandwidth, and are frequently employed in
radio-frequency
applications, for which a satisfactory coupling coefficient is
maintained by carefully
overlapping the primary and secondary windings. They're also
used for resonant
transformers such as Tesla coils where they can achieve
reasonably low loss in spite of
the high leakage inductance.
Windings
Figure 2.4 Windings are usually arranged concentrically to
minimize flux
leakage.
The conducting material used for the windings depends upon the
application,
but in all cases the individual turns must be electrically
insulated from each other to
ensure that the current travels throughout every turn.For small
power and signal
transformers, in which currents are low and the potential
difference between adjacent
turns are there.
Figure 2.5 Winding shapes
Cut view through transformer windings. White: insulator. Green
spiral: Grain oriented
silicon steel. Black: Primary winding made of oxygen-free
copper. Red: Secondary
winding. Top left: Toroidal transformer. Right: C-core, but
E-core would be similar.
The black windings are made of film. Top: Equally low
capacitance between all ends
of both windings. Since most cores are at least moderately
conductive they also need
insulation. Bottom: Lowest capacitance for one end of the
secondary winding needed
for low-power high-voltage transformers. Bottom left: Reduction
of leakage would
lead to increase of capacitance.
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Large power transformers use multiple-stranded conductors as
well, since even
at low power frequencies non-uniform distribution of current
would otherwise exist in
high-current windings. Each strand is individually insulated,
and the strands are
arranged so that at certain points in the winding, or throughout
the whole winding,
each portion occupies different relative positions in the
complete conductor. The
transposition equalizes the current flowing in each strand of
the conductor, and
reduces eddy current losses in the winding itself. The stranded
conductor is also more
flexible than a solid conductor of similar size, aiding
manufacture.
For signal transformers, the windings may be arranged in a way
to minimize
leakage inductance and stray capacitance to improve
high-frequency response. This
can be done by splitting up each coil into sections, and those
sections placed in layers
between the sections of the other winding. This is known as a
stacked type or
interleaved winding.
Power transformers often have internal connections or taps at
intermediate
points on the winding, usually on the higher voltage winding
side, for voltage
regulation control purposes. Such taps are normally manually
operated, automatic on-
load tap changers being reserved, for cost and reliability
considerations, to higher
power rated or specialized transformers supplying transmission
or distribution circuits
or certain utilization loads such as furnace transformers.
Audio-frequency
transformers, used for the distribution of audio to public
address loudspeakers, have
taps to allow adjustment of impedance to each speaker. A center
is often used in the
output stage of an audio power amplifier in a push-pull circuit.
Modulation
transformers in AM transmitters are very similar.Certain
transformers have the
windings protected by epoxy resin. By impregnating the
transformer with epoxy under
a vacuum, one can replace air spaces within the windings with
epoxy, thus sealing the
windings and helping to prevent the possible formation of corona
and absorption of
dirt or water. This produces transformers more suited to damp or
dirty environments,
but at increased manufacturing cost.
Cooling
Figure 2.6 Cooling
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Cutaway view of oil-filled power transformer. The conservator
(reservoir) at
top provides oil-to-atmosphere isolation. Tank walls' cooling
fins provide required
heat dissipation balance.
Though it is not uncommon for oil-filled transformers to have
today been in
operation for over fifty years high temperature damages winding
insulation, the
accepted rule of thumb being that transformer life expectancy is
halved for every 8
degree C increase in operating temperature. At the lower end of
the power rating
range, dry and liquid-immersed transformers are often
self-cooled by natural
convection and radiation heat dissipation. As power ratings
increase, transformers are
often cooled by such other means as forced-air cooling,
force-oil cooling, water-
cooling, or a combinations of these. The dialectic coolant used
in many outdoor utility
and industrial service transformers is transformer oil that both
cools and insulates the
windings. Transformer oil is a highly refined mineral oil that
inherently helps
thermally stabilize winding conductor insulation, typically
paper, within acceptable
insulation temperature rating limitations. However, the heat
removal problem is
central to all electrical apparatus such that in the case of
high value transformer assets,
this often translates in a need to monitor, model, forecast and
manage oil and winding
conductor insulation temperature conditions under varying,
possibly difficult, power
loading conditions. Indoor liquid-filled transformers are
required by building
regulations in many jurisdictions to either use a non-flammable
liquid or to be located
in fire-resistant rooms. Air-cooled dry transformers are
preferred for indoor
applications even at capacity ratings where oil-cooled
construction would be more
economical, because their cost is offset by the reduced building
construction cost.
The oil-filled tank often has radiators through which the oil
circulates by
natural convection. Some large transformers employ
electric-operated fans or pumps
for forced-air or forced-oil cooling or heat exchanger-based
water-cooling. Oil-filled
transformers undergo prolonged drying processes to ensure that
the transformer is
completely free of water before the cooling oil is introduced.
This helps prevent
electrical breakdown under load. Oil-filled transformers may be
equipped
with Buchholz relays, which detect gas evolved during internal
arcing and rapidly de-
energize the transformer to avert catastrophic failure.
Oil-filled transformers may fail,
rupture, and burn, causing power outages and losses.
Installations of oil-filled
transformers usually include fire protection measures such as
walls, oil containment,
and fire-suppression sprinkler systems.
Insulation drying
Construction of oil-filled transformers requires that the
insulation covering the
windings be thoroughly dried before the oil is introduced. There
are several different
methods of drying. Common for all is that they are carried out
in vacuum
environment. The vacuum makes it difficult to transfer energy
(heat) to the insulation.
For this there are several different methods. The traditional
drying is done by
circulating hot air over the active part and cycle this with
periods of hot-air vacuum
(HAV) drying. More common for larger transformers is to use
evaporated solvent
which condenses on the colder active part. The benefit is that
the entire process can be
carried out at lower pressure and without influence of added
oxygen. This process is
commonly called vapor-phase drying (VPD).
For distribution transformers, which are smaller and have a
smaller insulation
weight, resistance heating can be used. This is a method where
current is injected in
the windings to heat the insulation. The benefit is that the
heating can be controlled
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very well and it is energy efficient. The method is called
low-frequency heating (LFH)
since the current is injected at a much lower frequency than the
nominal of the grid,
which is normally 50 or 60 Hz. A lower frequency reduces the
effect of the inductance
in the transformer, so the voltage needed to induce the current
can be reduced. The
LFH drying method is also used for service of older
transformers.
Terminals
Very small transformers will have wire leads connected directly
to the ends of
the coils, and brought out to the base of the unit for circuit
connections. Larger
transformers may have heavy bolted terminals, bus bars or
high-voltage
insulated bushings made of polymers or porcelain. A large
bushing can be a complex
structure since it must provide careful control of the electric
field gradient without
letting the transformer leak oil.
2.1.2 An ideal Transformer
Figure 2.7 Basic principle of Operation
An ideal transformer. The secondary current arises from the
action of the
secondary EMF on the (not shown) load impedance.The transformer
is based on two
principles: first, that an electric current can produce a
magnetic
field (electromagnetism) and second that a changing magnetic
field within a coil of
wire induces a voltage across the ends of the coil
(electromagnetic induction).
Changing the current in the primary coil changes the magnetic
flux that is developed.
The changing magnetic flux induces a voltage in the secondary
coil.
An ideal transformer is shown in the adjacent figure. Current
passing through
the primary coil creates a magnetic field. The primary and
secondary coils are
wrapped around a core of very high magnetic, such as iron, so
that most of the
magnetic flux passes through both the primary and secondary
coils. If a load is
connected to the secondary winding, the load current and voltage
will be in the
directions indicated, given the primary current and voltage in
the directions indicated
(each will be alternating current in practice).
2.1.3 Induction Law
The voltage induced across the secondary coil may be
calculated
from Faraday's law of induction, which states that:
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where Vs is the instantaneous voltage, Ns is the number of turns
in the secondary coil
and is the magnetic flux through one turn of the coil. If the
turns of the coil are oriented perpendicularly to the magnetic
field lines, the flux is the product of
the magnetic flux density B and the area A through which it
cuts. The area is constant,
being equal to the cross-sectional area of the transformer core,
whereas the magnetic
field varies with time according to the excitation of the
primary. Since the same
magnetic flux passes through both the primary and secondary
coils in an ideal
transformer, the instantaneous voltage across the primary
winding equals
Taking the ratio of the two equations for Vs and Vp gives the
basic equation for
stepping up or stepping down the voltage
Np/Ns is known as the turns ratio, and is the primary functional
characteristic of any
transformer. In the case of step-up transformers, this may
sometimes be stated as the
reciprocal, Ns/Np. Turns ratio is commonly expressed as an
irreducible fraction or
ratio: for example, a transformer with primary and secondary
windings of,
respectively, 100 and 150 turns is said to have a turns ratio of
2:3 rather than 0.667 or
100:150.
An elementary transformer consists of a soft iron or silicon
steel core and two
windings, placed on it. The windings are insulated from both the
core and each other.
The core is built up of thin soft iron or low reluctance to the
magnetic flux. The
winding connected to the magnetic flux. The winding connected to
the supply main is
called the primary and the winding connected to the load circuit
is called the
secondary.
Although in the actual construction the two windings are usually
wound one
over the other, for the sake of simplicity, the figures for
analyzing transformer theory
show the windings on opposite sides of the core, as shown below
Simple Transformer
.
When primary winding is connected to an ac supply mains, current
flows
through it. Since this winding links with an iron core, so
current flowing through this
winding produces an alternating flux in the core. Since this
flux is alternating and links
with the secondary winding also, so induces an emf in the
secondary winding.
The frequency of induced emf in secondary winding is the same as
that of the
flux or that of the s supply voltage. The induced emf in the
secondary winding enables
it to deliver current to an external load connected across it.
Thus the energy is
transformed from primary winding to the secondary winding by
means of electro-
magnetic induction without anychange in frequency. The flux of
the iron core links
not only with the secondary winding but also with the primary
winding, so produces
self-induced emf in the primary winding:
This induced in the primary winding opposes the applied voltage
and therefore
sometimes it is known as back emf of the primary. In fact the
induced emf in the
primary winding limits the primary current in much the same way
that the back emf in
a dc motor limits the armature current.
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Transformation ratio.
The ratio of secondary voltage to primary voltage is known as
the voltage
transformation ratio and is designated by letter K. i.e. Voltage
transformation ratio.
Current ratio.
The ratio of secondary current to primary current is known as
current ratio and
is reciprocal of voltage transformation ratio in an ideal
transformer.
2.2 Equivalent Circuit The electrical circuit for any electrical
engineering device can be drawn if
theequations describing its behavior are known. The equivalent
circuit for
electromagneticdevice is a combination of resistances,
inductances, capacitances,
voltages etc. In theequivalent circuit, (R1+jX1) and (R2+jX2)
are the leakage
impedances of the primary andsecondary windings respectively.
The primary current
I1 consists of two components.One component, I1 is the load
component and the
second is no-load current Io which iscomposed of Ic and Im. The
current Ic is in phase
with E1 and the product of these twogives core loss. Ro
represents the core loss and is
called core-loss resistance. The currentIm is represented by a
reactance Xo and is
called magnetizing reactance. The transformermagnetization curve
is assumed linear,
since the effect of higher order harmonics cant berepresented in
the equivalent circuit. In transformer analysis, it is usual to
transfer thesecondary quantities to primary side
or primary quantities to secondary side.
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Figure 2.8 Equivalent Circuit
2.3 Transformer Losses
1. Primary copper loss
2. Secondary copper loss
3. Iron loss
4. Dielectric loss
5. Stray load loss
These are explained in sequence below.
Primary and secondary copper losses take place in the respective
winding resistances
due to the flow of the current in them. The primary and
secondary resistances differ
from their d.c. values due to skin effect and the temperature
rise of the windings.
While the average temperature rise can be approximately used,
the skin effect is harder
to get analytically. The short circuit test gives the value of
Re taking into account the
skin effect.
The iron losses contain two components - Hysteresis loss and
Eddy current
loss. The Hysteresis loss is a function of the material used for
the core.Ph = KhB1.6f
For constant voltage and constant frequency operation this can
be taken to be constant.
The eddy current loss in the core arises because of the induced
emf in the steel
lamination sheets and the eddies of current formed due to it.
This again producesa
power loss Pe in the lamination.wheret is the thickness of the
steel lamination used. As
the lamination thickness is much smaller than the depth of
penetration of the field, the
eddy current loss can be reduced by reducing the thickness of
the lamination. Present
day laminations are of 0.25 mm thickness and are capable of
operation at 2 Tesla.
These reduce the eddy current losses in the core.This loss also
remains constant
due to constant voltage and frequency of operation. The sum of
hysteresis and eddy
current losses can be obtained by the open circuit test.The
dielectric losses take place
in the insulation of the transformer due to the large electric
stress. In the case of low
voltage transformers this can be neglected. For constant voltage
operation this can be
assumed to be a constant. The stray load losses arise out of the
leakage fluxes of the
transformer. These leakage fluxes link the metallic structural
parts, tank etc. and
produce eddy current losses in them. Thus they take place all
round the transformer instead of a definite place , hence the name
stray. Also the leakage flux is directly proportional to the load
current unlike the mutual flux which is proportional to the
applied voltage. Hence this loss is called stray load loss.This
can also be estimated experimentally.
It can be modeled by another resistance in the series branch in
the equivalent
circuit. The stray load losses are very low in air-cored
transformers due to the absence
of the metallic tank. Thus, the different losses fall in to two
categories Constant losses
(mainly voltage dependant) and Variable losses (current
dependant). The expression
for the efficiency of the transformer operating at a fractional
load x of its rating, at a
load power factor of 2, can be written as losses and Pvar the
variable losses at full
load.For a given power factor an expression for in terms of the
variable x is thus
obtained.Bydifferentiating with respect to x and equating the
same to zero, the
condition formaximum efficiency is obtained. The maximum
efficiency it can be
easily deduced that thismaximum value increases with increase in
power factor and is
zero at zero power factor of the load. It may be considered a
good practice to select the
operating load point to be at the maximum efficiency point. Thus
if a transformer is on
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full load, for most part of the time then the max can be made to
occur at full load by
proper selection of constant and variablelosses.However, in the
modern transformers
the iron losses are so low that it is practicallyimpossible to
reduce the full load copper
losses to that value. Such a design wastes lot of copper. This
point is illustrated with
the help of an example below.Two 100 kVA transformers And B are
taken. Both
transformers have total full loadlosses to be 2 kW. The break up
of this loss is chosen
to be different for the two transformers.Transformer A: iron
loss 1 kW, and copper
loss is 1 kW. The maximum efficiency of 98.04%occurs at full
load at unity power
factor.Transformer B: Iron loss =0.3 kW and full load copper
loss =1.7 kW. This also
has a full load of 98.04%. Its maximum occurs at a fractional
load of q0.31.7 = 0.42.
The maximum efficiency at unity power factor being at the
corresponding point the
transformer A has an efficiency of Transformer A uses iron of
more loss per kg at a
given flux density, but transformer B uses lesser quantity of
copper and works at
higher current density.
When the primary of a transformer is connected to the source of
an ac supply
and the secondary is open circuited, the transformer is said to
be on no load. Which
will create alternating flux. No-load current, also known as
excitation or exciting
current has two components the magnetizing component Im and the
energy component
Ie.
Figure2.9 Transformer on No Load
Im is used to create the flux in the core and Ie is used to
overcome the
hysteresis and eddy current losses occurring in the core in
addition to small amount of
copper losses occurring in the primary only (no copper loss
occurs in the secondary,
because it carries no current, being open circuited.) From
vector diagram shown in
above it is obvious that
1. Induced emfs in primary and secondary windings, and lag the
main flux by and
are in phase with each other.
2. Applied voltage to primary and leads the main flux by and is
in phase opposition to
.
3. Secondary voltage is in phase and equal to since there is no
voltage drop in
secondary.
4. is in phase with and so lags
5. is in phase with the applied voltage .
6. Input power on no load = cos where
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Transformer on Load
The transformer is said to be loaded, when its secondary circuit
is completed
through an impedance or load. The magnitude and phase of
secondary current (i.e.
current flowing through secondary) with respect to secondary
terminals depends upon
the characteristic of the load i.e. current will be in phase,
lag behind and lead the
terminal voltage respectively when the load is non-inductive,
inductive and
capacitive. The net flux passing through the core remains almost
constant from no-
load to full load irrespective of load conditions and so core
losses remain almost
constant from no-load to full load.
Secondary windings Resistance and Leakage Reactance In actual
practice, both
of the primary and have got some ohmic resistance causing
voltage drops and copper
losses in the windings. In actual practice, the total flux
created does not link both of
the primary and secondary windings but is divided into three
components namely the
main or mutual flux linking both of the primary and secondary
windings, primary
leakage flux linking with primary winding only and secondary
leakage flux linking
with secondary winding only.
The primary leakage flux is produced by primary ampere-turns and
is
proportional to primary current, number of primary turns being
fixed. The primary
leakage flux is in phase with and produces self inducedemf is in
phase with and
produces self inducedemf E given as 2f in the primary winding.
The self inducedemf
divided by the primary current gives the reactance of primary
and is denoted by .
i.e. E = 2f
2.4 Transformer Tests
1 .Open-circuit or no-load test
2.Short circuit or impedance test
2.4.1 Open-circuit or No-load Test.
In this test secondary (usually high voltage) winding is left
open, all metering
instruments (ammeter, voltmeter and wattmeter) are connected on
primary side and
normal rated voltage is applied to the primary (low voltage)
winding, as illustrated
below
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Figure2.10 Open Circuit
Iron loss = Input power on no-load W0 watts (wattmeter reading)
No-load current = 0
amperes (ammeter reading) Angle of lag, = /Io Ie = and Im = o -
Caution: Since no load current I0 is very small, therefore,
pressure coils of watt meter and the volt meter
should be connected such that the current taken by them should
not flow through the
current taken by them should not flow through the current coil
of the watt meter.
2.4.2 Short-circuit or Impedance Test.
This test is performed to determine the full-load copper loss
and equivalent
resistance and reactance referred to secondary side. In this
test, the terminals of the
secondary (usually the low voltage) winding are short circuited,
all meters (ammeter,
voltmeter and wattmeter) are connected on primary side and a low
voltage, usually 5
to 10 % of normal rated primary voltage at normal frequency is
applied to the primary,
as shown in fig below.
The applied voltage to the primary, say Vs is gradually
increased till the ammeter A indicates the full load current of the
side in which it is connected. The
reading Ws of the wattmeter gives total copper loss (iron losses
being negligible due to
very low applied voltage resulting in very small flux linking
with the core) at full load.
Le the ammeter reading be Is.
Figure 2.11Short Circuit
Equivalent impedence referred to primary= Commercial Efficiency
and
Allday Efficiency (a) Commercial Efficiency. Commercial
efficiency is defined as the
ratio of power output to power input in kilowatts.(b) All-day
Efficiency. The all day
efficiency is defined as the ratio of output in kwh to the input
in kwh during the whole
day. Transformers used for distribution are connected for the
whole day to the line but
loaded intermittently. Thus the core losses occur for the whole
day but copper losses
occur only when the transformer is delivering the load current.
Hence if the
transformer is not used to supply the load current for the whole
day all day efficiency
will be less than commercial efficiency. The efficiency
(commercial efficiency) will
be maximum when variable losses (copper losses) are equal to
constant losses (iron or
core losses).sign is for inductive load and sign is for
capacitive load Transformer
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efficiency, Where x is the ratio of secondary current I2 and
rated full load secondary
current.
2.5 Efficiency
Transformers which are connected to the power supplies and loads
and are in
operation are required to handle load current and power as per
the requirements of the
load. An unloaded transformer draws only the magnetization
current on the primary
side, the secondary current being zero. As the load is increased
the primary and
secondary currents increase as per the load requirements. The
volt amperes and
wattage handled by the transformer also increases. Due to the
presence of no load
losses and I2R losses in the windings certain amount of
electrical energy gets
dissipated as heat inside the transformer.
This gives rise to the concept of efficiency. Efficiency of a
power equipment is
dened at any load as the ratio of the power output to the power
input. Putting in the form of an expression, while the efficiency
tells us the fraction of the input power
delivered to the load, the deficiency focuses our attention on
losses taking place inside
transformer. As a matter of fact the losses heat up machine. The
temperature rise
decides the rating of the equipment. The temperature rise of the
machine is a function
of heat generated the structural configuration, method of
cooling and type of loading
(or duty cycle of load). The peak temperature attained directly
affects the life of the
insulations of the machine for any class of insulation.
These aspects are briefly mentioned under section load test.The
losses that take
place inside the machine expressed as a fraction of the input is
sometimes termed as
deficiency. Except in the case of an ideal machine, a certain
fraction of the input
power gets lost inside the machine while handling the power.
Thus the value for the
efficiency is always less than one. In the case of a.c. machines
the rating is expressed
in terms of apparent power. It is nothing but the product of the
applied voltage and the
current drawn. The actual power delivered is a function of the
power factor at which
this current is drawn.
As the reactive power shuttles between the source and the load
and has a zero
average value over a cycle of the supply wave it does not have
any direct effect on the
efficiency. The reactive power however increases the current
handled by the machine
and the losses resulting from it. Therefore the losses that take
place inside a
transformer at any given load play a vital role in determining
the efficiency. The losses
taking place inside a transformer can be enumerated as
below:
1. Primary copper loss
2. Secondary copper loss
3. Iron loss
4. Dielectric loss
5. Stray load loss
These are explained in sequence below.
Primary and secondary copper losses take place in the respective
winding
resistancesdue to the flow of the current in them. The primary
and secondary
resistances differ from their d.c. values due to skin effect and
the temperature rise of
the windings. While the average temperature rise can be
approximately used, the skin
effect is harder to get analytically. The short circuit test
gives the value of Re taking
into account the skin effect.The iron losses contain two
components Hysteresis loss
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and Eddy current loss. The Hysteresis loss is a function of the
material used for the
core.Ph = KhB1.6f For constant voltage and constant frequency
operation this can be
taken to be constant. The eddy current loss in the core arises
because of the induced
emf in the steel lamination sheets and the eddies of current
formed due to it. This again
producesa power loss Pe in the lamination. Where t is the
thickness of the steel
lamination used. As the lamination thickness is much smaller
than the depth of
penetration of the field, the eddy current loss can be reduced
by reducing the thickness
of the lamination. Present day laminations are of 0.25 mm
thickness and are capable of
operation at 2 Tesla.
These reduce the eddy current losses in the core.This loss also
remains constant
due to constant voltage and frequency of operation. The sum of
hysteresis and eddy
current losses can be obtained by the open circuit test.The
dielectric losses take place
in the insulation of the transformer due to the large electric
stress. In the case of low
voltage transformers this can be neglected. For constant voltage
operation this can be
assumed to be a constant. The stray load losses arise out of the
leakage fluxes of the
transformer. These leakage fluxes link the metallic structural
parts, tank etc. and
produce eddy current losses in them. Thus they take place all
round the transformer instead of a definite place, hence the name
stray. Also the leakage flux is directly proportional to the load
current unlike the mutual flux which is proportional to the
applied voltage.
Hence this loss is called stray load loss.This can also be
estimated experimentally. It can be modeled by another resistance
in the series branch in the
equivalent circuit. The stray load losses are very low in
air-cored transformers due to
the absence of the metallic tank. Thus, the different losses
fall in to two categories
Constant losses (mainly voltage dependant) and Variable losses
(current dependant).
The expression for the efficiency of the transformer operating
at a fractional load x of
its rating, at a load power factor of 2 can be written as losses
and Pvar the variable
losses at full load. For a given power factor an expression for
_ in terms of the variable
x is thus obtained. By differentiating _ with respect to x and
equating the same to zero,
the condition for maximum efficiency is obtained. The maximum
efficiency it can be
easily deduced that this Maximum value increases with increase
in power factor and is
zero at zero power factor of the load. It may be considered a
good practice to select the
operating load point to be at the maximum efficiency point.
Thus if a transformer is on full load, for most part of the time
then the max can
be made to occur at full load by proper selection of constant
and
variablelosses.However, in the modern transformers the iron
losses are so low that it is
practically impossible to reduce the full load copper losses to
that value. Such a design
wastes lot of copper. This point is illustrated with the help of
an example below. Two
100 kVA transformers A and B are taken. Both transformers have
total full load losses
to be 2 kW. The breakup of this loss is chosen to be different
for the two transformers.
Transformer A: iron loss 1 kW, and copper loss is 1 kW. The
maximum efficiency of
98.04%occurs at full load at unity power factor. Transformer B:
Iron loss =0.3 kW and
full load copper loss =1.7 kW. This also has a full load of
98.04%. Its maximum
occurs at a fractional load of q0.31.7 = 0.42. The maximum
efficiency at unity power
factor being at the corresponding point the transformer A has an
efficiency of
Transformer A uses iron of more loss per kg at a given flux
density, but transformer B
uses lesser quantity of copper and works at higher current
density.
% Efficiency = 100
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All day efficiency
Large capacity transformers used in power systems are classified
broadly into Power
transformers and Distribution transformers. The former variety
is seen in generating
stations and large substations. Distribution transformers are
seen at the distribution
substations. The basic difference between the two types arises
from the fact that the
power transformers are switched in or out of the circuit
depending upon the load to be
handled by them. Thus at 50% load on the station only 50% of the
transformers need
to be connected in the circuit. On the other hand a distribution
transformer is never
switched off. It has to remain in the circuit irrespective of
the load connected. In such
cases the constant loss of the transformer continues to be
dissipated. Hence the
concept of energy based efficiency is defined for such
transformers. It is called all day efficiency. The all day
efficiency is thus the ratio of the energy output of the
transformer over a day to the corresponding energy input. One day
is taken as duration
of time over which the load pattern repeats itself. This
assumption, however, is far
from being true. The power output varies from zero to full load
depending on the
requirement of the user and the load losses vary as the square
of the fractional loads.
The no-load losses or constant losses occur throughout the 24
hours. Thus, the
comparison of loads on different days becomes difficult. Even
the load factor, which is
given by the ratio of the average load to rated load, does not
give satisfactory results.
The calculation of the all day efficiency is illustrated below
with an example. The
graph of load on the transformer, expressed as a fraction of the
full load is plotted
against time. In an actual situation the load on the transformer
continuously changes.
This has been presented by a stepped curve for convenience. For
the same load factor
different average loss can be there depending upon the values of
xi and ti. Hence a
better option would be to keep the constant losses very low to
keep the all day
efficiency high. Variable losses are related to load and are
associated with revenue
earned. The constant loss on the other hand has to be incurred
to make the service
available. The concept of all day efficiency may therefore be
more useful for
comparing two transformers subjected to the same load cycle. The
concept of
minimizing the lost energy comes into effect right from the time
of procurement of the
transformer. The constant losses and variable losses are
capitalized and added to the
material cost of the transformer in order to select the most
competitive one, which
gives minimum cost taking initial cost and running cost put
together. Obviously the
iron losses are capitalized more in the process to give an
effect to the maximization of
energy efficiency. If the load cycle is known at this stage, it
can also be incorporated
in computation of the best transformer.
2.6 Voltage Regulation
With the increase in load on the transformer, there is a change
in its terminal
voltage. The voltage falls if the load power factor is lagging.
It increases if power is
leading. The change in secondary terminal voltage from full load
to no load, expressed
as a percentage of full load voltage is called the percentage
voltage regulation of the
transformer
% Regulation E- V/V x 100.
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2.6.1 Circuit Diagram
Figure 2.2Load Test
2.6.2 Procedure:
Connect the circuit diagram as shown in fig (a) Apply full load
and note down the readings of wattmeter, voltmeter and
ammeter.
Decrease the load and note down the readings. Calculate
efficiency and regulation.
2.6.3 Observation Table
Wl Vl I2
2.6.4 Calculation
= V2 I2 / Wi * 100
% Reg = E V * 100 / V
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